-------------------------- MODULE MemorySafetyProof ---------------------------

EXTENDS Naturals, Sequences, FiniteSets
IMPORT MemoryManager

(*
 * 内存管理器安全属性的形式化证明
 * 
 * 本文件包含ZENrs exokernel中内存管理器安全属性的形式化证明，
 * 特别是关于"已分配的内存块不重叠"这一关键安全属性的证明。
 *)

(* 定义一些辅助定理 *)

(* 定理1：如果两个内存块在分配后不重叠，那么在任何后续操作后它们仍然不重叠 *)
Theorem PreserveNonOverlapAfterOperations == 
    ASSUME 
        (* 假设初始状态下没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks,
        
        (* 假设执行一个操作 *)
        MemoryManager!Next(mms, mms')
    PROVE 
        (* 证明执行操作后仍然没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks'

(* 定理2：新分配的内存块不会与任何已有的已分配内存块重叠 *)
Theorem NewAllocatedBlockDoesNotOverlapExisting == 
    ASSUME 
        (* 假设初始状态下没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks,
        
        (* 假设执行内存分配操作 *)
        (\E size, alignment \in MemoryManager!MemorySize : 
            mms' = MemoryManager!AllocatePhysicalMemory(mms, size, alignment)
        )
    PROVE 
        (* 证明执行分配后仍然没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks'

(* 定理3：释放内存块不会导致其他已分配块重叠 *)
Theorem FreeingBlockDoesNotCauseOverlap == 
    ASSUME 
        (* 假设初始状态下没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks,
        
        (* 假设执行内存释放操作 *)
        (\E addr \in MemoryManager!PhysicalAddress, size \in MemoryManager!MemorySize : 
            mms' = MemoryManager!FreePhysicalMemory(mms, addr, size)
        )
    PROVE 
        (* 证明执行释放后仍然没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks'

(* 定理4：内存映射操作不会导致已分配块重叠 *)
Theorem MappingDoesNotCauseBlockOverlap == 
    ASSUME 
        (* 假设初始状态下没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks,
        
        (* 假设执行内存映射操作 *)
        (\E virt_addr \in MemoryManager!VirtualAddress, phys_addr \in MemoryManager!PhysicalAddress, size \in MemoryManager!MemorySize, perms \in MemoryManager!MemoryPermissions : 
            mms' = MemoryManager!MapMemory(mms, virt_addr, phys_addr, size, perms)
        )
    PROVE 
        (* 证明执行映射后仍然没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks'

(* 定理5：内存解除映射操作不会导致已分配块重叠 *)
Theorem UnmappingDoesNotCauseBlockOverlap == 
    ASSUME 
        (* 假设初始状态下没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks,
        
        (* 假设执行内存解除映射操作 *)
        (\E virt_addr \in MemoryManager!VirtualAddress, size \in MemoryManager!MemorySize : 
            mms' = MemoryManager!UnmapMemory(mms, virt_addr, size)
        )
    PROVE 
        (* 证明执行解除映射后仍然没有重叠的已分配块 *)
        MemoryManager!NoOverlappingAllocatedBlocks'

(* 主定理：内存管理器始终保持已分配块不重叠的属性 *)
MainTheorem MemoryManagerPreservesNonOverlappingProperty == 
    MemoryManager!Spec => []MemoryManager!NoOverlappingAllocatedBlocks

(* 主定理的证明草图 *)
(* 1. 归纳法基础：初始状态满足NoOverlappingAllocatedBlocks *)
(* 2. 归纳步骤：假设当前状态满足NoOverlappingAllocatedBlocks，证明下一个状态也满足 *)
(* 3. 由Theorem PreserveNonOverlapAfterOperations，所有操作都保持该属性 *)
(* 4. 因此，所有可达状态都满足NoOverlappingAllocatedBlocks *)

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